Find minimum of the objective function F(a,b)=7a+18b if the feasible region is given by constraints a≥0, b≥0, 4a+6b≥24, and 2a+5b≥16.
To find the minimum of the objective function F(a, b) = 7a + 18b, given the constraints a ≥ 0, b ≥ 0, 4a + 6b ≥ 24, and 2a + 5b ≥ 16, we can use the method of linear programming.
Step 1: Plot the feasible region:
- Start by graphing the lines representing the constraints.
- First, graph the lines 4a + 6b = 24 and 2a + 5b = 16 on a coordinate plane.
- To graph these lines, you can find two points on each line by choosing any values for a or b and solving for the other variable.
- Then draw the lines through the points.
Step 2: Determine the feasible region:
- The feasible region is the region on the coordinate plane that satisfies all the constraints.
- Shade the region that is above or on the line 4a + 6b = 24 and above or on the line 2a + 5b = 16.
Step 3: Identify the corners of the feasible region:
- The minimum or maximum point of the objective function occurs at one of the corners of the feasible region.
- Locate the corners or vertices of the shaded region in the graph.
Step 4: Evaluate the objective function at each corner point:
- Substitute the x and y coordinates of each corner point into the objective function F(a, b) = 7a + 18b.
- Calculate the result for each corner point.
Step 5: Find the minimum objective function value:
- Identify the smallest value obtained from step 4. This will be the minimum value of the objective function F(a, b).
Therefore, following these steps, you can find the minimum value of the objective function F(a, b) = 7a + 18b for the given feasible region and constraints.
Just check the value of F at the vertices of the region, which are at
(0,16/5), (6,0), (3,2)